Angular Velocity of Merry-go-round

In summary, a 36 kg runner running at 2.9 m/s jumps onto a merry-go-round with a moment of inertia of 404 kgm^2. After impact, the merry-go-round and runner have a combined moment of inertia of 576 kgm^2. The final angular velocity of the merry-go-round is 0.381 rad/s.
  • #1
U154756
9
0
The problem reads as follows:

A runner of mass m=36 kg and running at 2.9 m/s runs and jumps on the rim of a playground merry-go-round which has a moment of inertia of 404 kgm^2 and a radius of 2 m. Assuming the merry-go-round is initially at rest, what is its final angular velocity to three decimal places?

According to the back of the book, the answer is 0.381 rad/s; however, I can never come up with that answer.

I have tried the following formulas with no luck:
mvr = Iw
1/2mv^2 = 1/2Iw^2 (until I realized kinetic energy would not be conserved.)

I know it has something to do with conservation of angular momentum; however, I cannot seem to figure out the angular momentum of the runner before he jumps onto the merry-go-round. Any help would be greatly appreciated.
 
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  • #2
U154756 said:
I know it has something to do with conservation of angular momentum; however, I cannot seem to figure out the angular momentum of the runner before he jumps onto the merry-go-round.
[tex]\vec{L} = \vec{r} \times \vec{p}[/tex]
 
  • #3
U154756 said:
I have tried the following formulas with no luck:
mvr = Iw
Actually, this is the one you need. The initial angular momentum (mvr) equals the final angular momentum. You'll need to find the rotational inertia (I) of the system (merry-go-round + runner).
 
  • #4
The angular momentum is the key. You are right about that. I am assuming the runner is approaching the merry-go-round on a tagent line. His angular velocity at the moment of jumping on is his speed times the radius of the merry-go-round, vr. His angular momentum is mvr. That angular momentum will be conserved.

The moment of inertia of the runner plus the merry-go-round after impact is the sum of the merry-go-round plus the moment of inertia of the runner, which is mr^2.

Can you take it from there?
 
  • #5
If L = r * p then I have already tried this as it would be the same as L = mvr. I have tried mvr = Iw and solved for w and got the formula w = mvr/I. When I plug the numbers into this formula I do not get the correct answer. Am I missing something?
 
  • #6
U154756 said:
Am I missing something?
What are you using for the moment of inertia?
 
  • #7
I finally got the correct answer. I hate it when you are on the right track, but can't find the missing part of the equation. My missing part of the equation was the moment of inertia of the runner. After using the formula mvr = (Imgr + Ir)w (where Imgr is inertia of merry-go-round and Ir is inertia of runner), I solved for w and was able to finally come up with the correct answer.

Thanks for your help.
 

Related to Angular Velocity of Merry-go-round

What is angular velocity?

Angular velocity is a measure of how fast an object is rotating around a fixed point, such as the center of a merry-go-round. It is typically measured in radians per second.

How is angular velocity different from linear velocity?

Angular velocity is a measure of the rate of change of angular displacement, while linear velocity is a measure of the rate of change of linear displacement. In other words, angular velocity measures how fast an object is rotating, while linear velocity measures how fast an object is moving in a straight line.

How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angular displacement by the change in time. It is typically represented by the symbol "ω" and is measured in radians per second.

How does the angular velocity of a merry-go-round affect the riders?

The angular velocity of a merry-go-round determines how fast it rotates, which in turn affects the speed and direction of the riders. A higher angular velocity will result in faster and more forceful spinning, while a lower angular velocity will result in slower and gentler spinning.

What factors can affect the angular velocity of a merry-go-round?

The angular velocity of a merry-go-round can be affected by several factors, including the amount of force applied to rotate it, the mass and distribution of weight of the riders, and any external forces such as friction or air resistance. Additionally, the shape and size of the merry-go-round can also impact its angular velocity.

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